Clustering Color/Intensity

1
Clustering Color/Intensity
Group together pixels of similar color/intensity.
2
Agglomerative Clustering

• Cluster connected pixels with similar color.
• Optimal decomposition may be hard.
• For example, find k connected components of image
  with least color variation.
• Greedy algorithm to make this fast.

3
Clustering Algorithm

• Initialize Each pixel is a region with color of
  that pixel and neighbors neighboring pixels.
• Loop
• Find adjacent two regions with most similar
  color.
• Merge to form new region with
• all pixels of these regions
• average color of these regions.
• All neighbors of either region.
• Stopping condition
• No regions similar
• Find k regions.

4
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
5
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
6
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
23 25 19 21 23
18 22 24.5 24.5 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
7
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
23 25 19 21 23
18 22 24.33 24.33 24.33
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
8
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
23 25 19 21 23
18 22 24.33 24.33 24.33
19.5 19.5 26 28 22
3 3 7 8 26
1 3 5 4 24
9
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
23 25 19 21 23
18 22 24.33 24.33 24.33
19.5 19.5 26 28 22
3 3 7.5 7.5 26
1 3 5 4 24
10
Example

11
Example
23 25 19 21 23
18 22 24 25 24
20 19 26 28 22
3 3 7 8 26
1 3 5 4 24
22.9 22.9 22.9 22.9 22.9
22.9 22.9 22.9 22.9 22.9
22.9 22.9 22.9 22.9 22.9
4.25 4.25 4.25 4.25 22.9
4.25 4.25 4.25 4.25 22.9
12
Clustering complexity

• n pixels.
• Initializing
• O(n) time to compute regions.
• Loop
• O(n) time to find closest neighbors (could speed
  up).
• O(n) time to update distance to all neighbors.
• At most n times through loop so O(nn) time
  total.

13
Agglomerative Clustering Discussion

• Start with definition of good clusters.
• Simple initialization.
• Greedy take steps that seem to most improve
  clustering.
• This is a very general, reasonable strategy.
• Can be applied to almost any problem.
• But, not guaranteed to produce good quality
  answer.

14
Parametric Clustering

• Each cluster has a mean color/intensity, and a
  radius of possible colors.
• For intensity, this is just dividing histogram
  into regions.
• For color, like grouping 3D points into spheres.

15
K-means clustering

• Brute force difficult because many spheres, many
  pixels.
• Assume all spheres same radius just need sphere
  centers.
• Iterative method.
• If we knew centers, it would be easy to assign
  pixels to clusters.
• If we knew which pixels in each cluster, it would
  be easy to find centers.
• So guess centers, assign pixels to clusters, pick
  centers for clusters, assign pixels to clusters,
  .
• matlab

16
K-means Algorithm

• Initialize Pick k random cluster centers
• Pick centers near data. Heuristics uniform
  distribution in range of data randomly select
  data points.
• Assign each point to nearest center.
• Make each center average of pts assigned to it.
• Go to step 2.

17
Lets consider a simple example. Suppose we want
to cluster black and white intensities, and we
have the intensities 1 3 8 11. Suppose we start
with centers c1 7 and c210. We assign 1, 3, 8
to c1, 11 to c2. Then we update c1 (138)/3
4, c2 11. Then we assign 1,3 to c1 and 8 and
11 to c2. Then we update c1 2, c2 9 ½. Then
the algorithm has converged. No assignments
change, so the centers dont change.
18
K-means Properties

• We can think of this as trying to find the
  optimal solution to
• Given points p1 pn, find centers c1ck
• and find mapping fp1pn-gtc1ck
• that minimizes C (p1-f(p1))2
  (pn-f(pn))2.
• Every step reduces C.
• The mean is the pt that minimizes sum of squared
  distance to a set of points. So changing the
  center to be the mean reduces this distance.
• When we reassign a point to a closer center, we
  reduce its distance to its cluster center.
• Convergence since there are only a finite set of
  possible assignments.

19
Local Minima

• However, algorithm might not find the best
  possible assignments and centers.
• Consider points 0, 20, 32.
• K-means can converge to centers at 10, 32.
• Or to centers at 0, 26.

20
E-M

• Like K-means with soft assignment.
• Assign point partly to all clusters based on
  probability it belongs to each.
• Compute weighted averages (cj) and variance (s).

Cluster centers are cj.
21
Example

• Matlab tutorial2
• Fuzzy assignment allows cluster to creep towards
  nearby points and capture them.

22
E-M/K-Means domains

• Used color/intensity as example.
• But same methods can be applied whenever a group
  is described by parameters and distances.
• Lines (circles, ellipses) independent motions
  textures (a little harder).